Add Josephus Problem

maths/josephus_problem.py

@@ -0,0 +1,91 @@
+"""
+The Josephus problem is a famous theoretical problem related to a certain
+counting-out game. This module provides functions to solve the Josephus problem
+for n people and a step size of k.
+
+The Josephus problem is defined as follows:
+- n people are standing in a circle.
+- Starting with a specified person, you count around the circle,
+  skipping a fixed number of people (k).
+- The person at which you stop counting is eliminated from the circle.
+- The counting continues until only one person remains.
+
+For more information about the Josephus problem, refer to:
+https://en.wikipedia.org/wiki/Josephus_problem
+"""
+
+
+def josephus_recursive(n: int, k: int) -> int:
+    """
+    Solve the Josephus problem for n people and a step size of k recursively.
+
+    Args:
+        n (int): Number of people.
+        k (int): Step size for elimination.
+
+    Returns:
+        int: The position of the last person remaining.
+
+    Examples:
+        >>> josephus_recursive(7, 3)
+        3
+        >>> josephus_recursive(10, 2)
+        4
+    """
+    if n == 1:
+        return 0
+
+    return (josephus_recursive(n - 1, k) + k) % n
+
+
+def find_winner(n: int, k: int) -> int:
+    """
+    Find the winner of the Josephus problem for n people and a step size of k.
+
+    Args:
+        n (int): Number of people.
+        k (int): Step size for elimination.
+
+    Returns:
+        int: The position of the last person remaining (1-based index).
+
+    Examples:
+        >>> find_winner(7, 3)
+        4
+        >>> find_winner(10, 2)
+        5
+    """
+    return josephus_recursive(n, k) + 1
+
+
+def josephus_iterative(n: int, k: int) -> int:
+    """
+    Solve the Josephus problem for n people and a step size of k iteratively.
+
+    Args:
+        n (int): The number of people in the circle.
+        k (int): The number of steps to take before eliminating someone.
+
+    Returns:
+        int: The position of the last person standing.
+
+    Examples:
+        >>> josephus_iterative(5, 2)
+        3
+        >>> josephus_iterative(7, 3)
+        4
+    """
+    circle = list(range(1, n + 1))
+    current = 0
+
+    while len(circle) > 1:
+        current = (current + k - 1) % len(circle)
+        circle.pop(current)
+
+    return circle[0]
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()